Heat the oil in a Dutch oven or large soup pot over medium heat until shimmering.
Add the diced onion and sauté until beginning to soften, about 5 minutes.
Stir in the cumin, oregano, kosher salt, cayenne, and minced garlic and cook for about 30 seconds until fragrant.
Add the chicken breasts, drained beans, diced green chiles, and chicken broth; bring to a boil over high heat.
Reduce heat to maintain a gentle, steady simmer and cook until the chicken reaches 155°F on an instant-read thermometer, about 12 to 14 minutes, turning the breasts once halfway through if they are not fully submerged.
Remove the chicken to a cutting board and let rest 10 minutes, then dice or shred into bite-sized pieces.
Use an immersion blender to puree a portion of the chili in the pot to thicken it while leaving some beans whole; alternately pulse a few ladles in a blender or food processor and return to the pot. Simmer 5 minutes more.
Stir the diced or shredded chicken, chopped cilantro, and lime juice into the chili and simmer 5 minutes to meld flavors.
Ladle into bowls, serve with optional toppings and an extra squeeze of lime, and enjoy.